The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data.
$\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius.
-Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector $\vec{r}$)
+Writing down the derivative of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$)
\begin{equation}
\frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big)
\label{eq:lj-d}
\end{equation}
one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$.
Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth.
-The $i$th component of the force $F^j$ on particle $j$ is obtained by
+The $i$th component of the force is given by
\begin{equation}
-F_i^j = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.}
+F_i = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.}
\label{eq:lj-f}
\end{equation}
\subsubsection{The Tersoff potential}
-Ther Tersoff potential \cite{tersoff1} \ldots
-
+Tersoff proposed an empirical interatomic potential for covalent systems.
+The Tersoff potential explicitly incorporates the dependence of bond order on local envirenments, permitting an improved description of covalent materials.
+Tersoff applied the potential to silicon \cite{tersoff_silicon1,tersoff_silicon2,tersoff_silicon3}, carbon \cite{tersoff_carbon} and also to multicomponent systems like $SiC$ \cite{tersoff_multi}.
+The basic idea is that, in real systems, the bond order depends upon the local environment.
+An atom with many neighbours forms weaker bonds than an atom with few neighbours.
+
+The interatomic potential is taken to have the form
+\begin{eqnarray}
+E & = & \sum_i E_i = \frac{1}{2} \sum_{i \ne j} V_{ij} \textrm{ ,} \\
+V_{ij} & = & f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ] \textrm{ .}
+\end{eqnarray}
+$E$ is the total energy of the system, constituted either by the sum over the site energies $E_i$ or by the bond energies $V_{ij}$.
+The indices $i$ and $j$ correspond to the atoms of the system with $r_{ij}$ being the distance from atom $i$ to atom $j$.
+
+The functions $f_R$ and $f_A$ represent a repulsive and an attractive pair potential.
+The repulsive part is due to the orthogonalization energy of overlapped atomic wave functions.
+The attractive part is associated with the bonding.
+\begin{eqnarray}
+f_R(r_{ij}) & = & A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
+f_A(r_{ij}) & = & -B_{ij} \exp (- \mu_{ij} r_{ij} )
+\end{eqnarray}
+The function $f_C$ is the potential cutoff function to limit the range of the potential.
+It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.
\begin{equation}
-V_{ij} = f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ]
+f_C(r_{ij}) = \left\{
+ \begin{array}{ll}
+ 1, & r_{ij} < R_{ij} \\
+ \frac{1}{2} + \frac{1}{2} \cos \Big[ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) \Big], & R_{ij} < r_{ij} < S_{ij} \\
+ 0, & r_{ij} > S_{ij}
+ \end{array} \right.
\end{equation}
-The total energy is then given by
-\begin{equation}
-E = \frac{1}{2} \sum_{i \ne j} V_{ij} \, \textrm{.}
-\end{equation}
+The function $b_{ij}$ represents a measure of the bond order, monotonically decreasing with the coordination of atoms $i$ and $j$.
+It is of the form:
+\begin{eqnarray}
+b_{ij} & = & \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i} \\
+\zeta_{ij} & = & \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk}) \\
+g(\theta_{ijk}) & = & 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] \\
+\end{eqnarray}
+where $\theta_{ijk}$ is the bond angle between bonds $ij$ and $ik$.
+This is illustrated in Figure \ref{img:tersoff_angle}.
+\printimg{!h}{width=8cm}{tersoff_angle.eps}{Angle between bonds of atoms $i,j$ and $i,k$.}{img:tersoff_angle}
-\begin{equation}
-f_R(r_{ij}) = A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
-\end{equation}
+Here comes an explanation, energy per bond monotonically decreasing with the amount of bonds and so on and so on \ldots
+The force acting on atom $i$ is given by the derivative of the potential energy.
+For a three body potential ($V_{ij} \neq V{ji}$) the derivation is of the form
\begin{equation}
-f_A(r_{ij}) = -B_{ij} \exp (- \mu_{ij} r_{ij} ) \\
+\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .}
\end{equation}
-
-The function $f_C$ is the potential cutoff function designed to have a smooth transition between $R_{ij}$ and $S_{ij}$.
+The force is then given by
\begin{equation}
-f_C(r_{ij}) = \left\{
- \begin{array}{ll}
- 1 & r_{ij} < R_{ij} \\
- \frac{1}{2} + \frac{1}{2} \cos [ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) ] & R_{ij} < r_{ij} < S_{ij} \\
- 0 & r_{ij} > S_{ij}
- \end{array} \right.
+F^i = - \nabla_{{\bf r}_i} E \textrm{ .}
\end{equation}
+The details of the Tersoff potential derivative can be seen in appendix \ref{app:d_tersoff}.
+And here comes why we use it. Advantages and disadvantages compared to other interaction potentials, maybe this is best at the very end of all potentials \ldots
\subsubsection{The Brenner potential}
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